Problem: $M(t)$ models the distance (in millions of $\text{km}$ ) from Mars to the Sun $t$ days after it's at its furthest point. Here, $t$ is entered in radians. $M(t) = 21\cos\left(\dfrac{2\pi}{687}t\right) + 228$ How many days later does Mars first reach $220$ million $\text{km}$ from the Sun? Round your final answer to the nearest whole day.
Solution: Converting the problem into mathematical terms $M(t) = 21\cos\left({\dfrac{2\pi}{687}}t\right) + 228$ has a period of $\dfrac{2\pi}{{\scriptsize\dfrac{2\pi}{687}}}=687$ days. We want to find the first solution to the equation $M(t)=220$ within the period $0<t<687$. The answer The equation's two solutions within the desired period (rounded to the nearest whole day) are $214$ and $473$. Therefore, Mars first reaches a distance of $220$ million $\text{km}$ from the Sun after about $214$ days.